Problem: What common fraction is exactly half-way between $\frac{2}{3}$ and $\frac{4}{5}$?
Answer: The average of two numbers is exactly half-way between them.  Therefore, $\frac{1}{2}\left(\frac{2}{3}+\frac{4}{5}\right)=\boxed{\frac{11}{15}}$ is half-way between $\frac{2}{3}$ and $\frac{4}{5}$.